Cumulative frequencies for 100 people: : 8, : 30, : 65, : 90, : 100.
- (a)How many people had a value greater than 40?[1]
- (b)Which class contains the median?[1]
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For grouped data you cannot find the exact median, but a lets you estimate it — along with the quartiles, the interquartile range and any percentile. Those five key values then summarise the data in a , the clearest way to compare two distributions side by side.
The big picture
Cumulative frequency is a running total: how many values are less than or equal to each upper class boundary. Plotting it produces an S-shaped curve, and reading across from , and gives the median and quartiles.
The interquartile range measures the spread of the middle half of the data, so it ignores extreme values — which is why Edexcel prefers it to the range for comparisons. A full comparison states which group has the higher median and which has the smaller IQR, each in context.
What you'll learn
Add frequencies as you go down the table to get the running total. The final cumulative frequency equals the total number of values.
Plot each cumulative frequency at the of its class, starting from zero at the lower boundary of the first class.
Join the points with a smooth curve or straight lines.
Class frequencies are 5, 12, 20, 9 and 4. What is the final cumulative frequency?
Tip — Plotting at midpoints is the classic cumulative frequency error. The running total counts everything up to the end of the class, so plot at the upper boundary.
Median: read across from on the cumulative frequency axis to the curve, then down to the value.
Lower quartile at ; upper quartile at . Interquartile range .
To find how many values are above or below a given value, read up from that value and across to the cumulative frequency.
A data set has lower quartile 18 and upper quartile 31. Find the interquartile range.
Reading values from a curve is an estimate, because the curve assumes values are spread smoothly within each class. Mark schemes accept a small range of readings.
A box plot shows five values on a scale: minimum, , median, and maximum.
The box runs from to with a line at the median; the whiskers extend to the minimum and maximum.
Box plots drawn on the same scale make comparisons immediate.
A box plot has minimum 12 and maximum 58. What is the range?
Tip — Draw box plots accurately against a scale. A box plot without a scale cannot be read.
Compare the to say which group is higher on average, and the (or ranges) to say which is more consistent.
Write each comparison in context, with the values.
Think like an examiner
Cumulative frequency
Watch out for these
Stretch yourself
Using the 80 runners’ data, estimate how many runners took longer than 38 minutes, assuming values are spread evenly within classes.
Hint — Find how many finished in under 38 minutes by working within the class.
Questions students ask
Key takeaways
Exam-style questions
3 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.
Cumulative frequencies for 100 people: : 8, : 30, : 65, : 90, : 100.
A box plot shows minimum 15, lower quartile 22, median 30, upper quartile 41, maximum 55.
Unlimited practice
A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.
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Test yourself
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Real past-paper questions on Cumulative Frequency & Box Plots, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.